(*  Title:      HOL/HOLCF/Tools/fixrec.ML

    Author:     Amber Telfer and Brian Huffman

Recursive function definition package for HOLCF.

*)

signature FIXREC =

sig

  val add_fixrec: (binding * typ option * mixfix) list

    -> (bool * (Attrib.binding * term)) list -> local_theory -> local_theory

  val add_fixrec_cmd: (binding * string option * mixfix) list

    -> (bool * (Attrib.binding * string)) list -> local_theory -> local_theory

  val add_matchers: (string * string) list -> theory -> theory

  val fixrec_simp_tac: Proof.context -> int -> tactic

  val setup: theory -> theory

end

structure Fixrec : FIXREC =

struct

open HOLCF_Library

infixr 6 ->>

infix -->>

infix 9 `

val def_cont_fix_eq = @{thm def_cont_fix_eq}

val def_cont_fix_ind = @{thm def_cont_fix_ind}

fun fixrec_err s = error ("fixrec definition error:\n" ^ s)

(*************************************************************************)

(***************************** building types ****************************)

(*************************************************************************)

local

fun binder_cfun (Type(@{type_name cfun},[T, U])) = T :: binder_cfun U

  | binder_cfun (Type(@{type_name "fun"},[T, U])) = T :: binder_cfun U

  | binder_cfun _   =  []

fun body_cfun (Type(@{type_name cfun},[_, U])) = body_cfun U

  | body_cfun (Type(@{type_name "fun"},[_, U])) = body_cfun U

  | body_cfun T   =  T

in

fun matcherT (T, U) =

  body_cfun T ->> (binder_cfun T -->> U) ->> U

end

(*************************************************************************)

(***************************** building terms ****************************)

(*************************************************************************)

val mk_trp = HOLogic.mk_Trueprop

(* splits a cterm into the right and lefthand sides of equality *)

fun dest_eqs t = HOLogic.dest_eq (HOLogic.dest_Trueprop t)

(* similar to Thm.head_of, but for continuous application *)

fun chead_of (Const(@{const_name Rep_cfun},_)$f$_) = chead_of f

  | chead_of u = u

infix 1 === val (op ===) = HOLogic.mk_eq

fun mk_mplus (t, u) =

  let val mT = Term.fastype_of t

  in Const(@{const_name Fixrec.mplus}, mT ->> mT ->> mT) ` t ` u end

fun mk_run t =

  let

    val mT = Term.fastype_of t

    val T = dest_matchT mT

    val run = Const(@{const_name Fixrec.run}, mT ->> T)

  in

    case t of

      Const(@{const_name Rep_cfun}, _) $

        Const(@{const_name Fixrec.succeed}, _) $ u => u

    | _ => run ` t

  end

(*************************************************************************)

(************* fixed-point definitions and unfolding theorems ************)

(*************************************************************************)

structure FixrecUnfoldData = Generic_Data

(

  type T = thm Symtab.table

  val empty = Symtab.empty

  val extend = I

  fun merge data : T = Symtab.merge (K true) data

)

local

fun name_of (Const (n, _)) = n

  | name_of (Free (n, _)) = n

  | name_of t = raise TERM ("Fixrec.add_unfold: lhs not a constant", [t])

val lhs_name =

  name_of o head_of o fst o HOLogic.dest_eq o HOLogic.dest_Trueprop o prop_of

in

val add_unfold : attribute =

  Thm.declaration_attribute

    (fn th => FixrecUnfoldData.map (Symtab.insert (K true) (lhs_name th, th)))

end

fun add_fixdefs

  (fixes : ((binding * typ) * mixfix) list)

  (spec : (Attrib.binding * term) list)

  (lthy : local_theory) =

  let

    val thy = Proof_Context.theory_of lthy

    val names = map (Binding.name_of o fst o fst) fixes

    val all_names = space_implode "_" names

    val (lhss, rhss) = ListPair.unzip (map (dest_eqs o snd) spec)

    val functional = lambda_tuple lhss (mk_tuple rhss)

    val fixpoint = mk_fix (mk_cabs functional)

    val cont_thm =

      let

        val prop = mk_trp (mk_cont functional)

        fun err _ = error (

          "Continuity proof failed please check that cont2cont rules\n" ^

          "or simp rules are configured for all non-HOLCF constants.\n" ^

          "The error occurred for the goal statement:\n" ^

          Syntax.string_of_term lthy prop)

        val rules = Cont2ContData.get lthy

        val fast_tac = SOLVED' (REPEAT_ALL_NEW (match_tac rules))

        val slow_tac = SOLVED' (simp_tac (simpset_of lthy))

        val tac = fast_tac 1 ORELSE slow_tac 1 ORELSE err

      in

        Goal.prove lthy [] [] prop (K tac)

      end

    fun one_def (Free(n,_)) r =

          let val b = Long_Name.base_name n

          in ((Binding.name (Thm.def_name b), []), r) end

      | one_def _ _ = fixrec_err "fixdefs: lhs not of correct form"

    fun defs [] _ = []

      | defs (l::[]) r = [one_def l r]

      | defs (l::ls) r = one_def l (mk_fst r) :: defs ls (mk_snd r)

    val fixdefs = defs lhss fixpoint

    val (fixdef_thms : (term * (string * thm)) list, lthy) = lthy

      |> fold_map Local_Theory.define (map (apfst fst) fixes ~~ fixdefs)

    fun pair_equalI (thm1, thm2) = @{thm Pair_equalI} OF [thm1, thm2]

    val tuple_fixdef_thm = foldr1 pair_equalI (map (snd o snd) fixdef_thms)

    val P = Var (("P", 0), map Term.fastype_of lhss ---> HOLogic.boolT)

    val predicate = lambda_tuple lhss (list_comb (P, lhss))

    val tuple_induct_thm = (def_cont_fix_ind OF [tuple_fixdef_thm, cont_thm])

      |> Drule.instantiate' [] [SOME (Thm.cterm_of thy predicate)]

      |> Local_Defs.unfold lthy @{thms split_paired_all split_conv split_strict}

    val tuple_unfold_thm = (def_cont_fix_eq OF [tuple_fixdef_thm, cont_thm])

      |> Local_Defs.unfold lthy @{thms split_conv}

    fun unfolds [] _ = []

      | unfolds (n::[]) thm = [(n, thm)]

      | unfolds (n::ns) thm = let

          val thmL = thm RS @{thm Pair_eqD1}

          val thmR = thm RS @{thm Pair_eqD2}

        in (n, thmL) :: unfolds ns thmR end

    val unfold_thms = unfolds names tuple_unfold_thm

    val induct_note : Attrib.binding * Thm.thm list =

      let

        val thm_name = Binding.qualify true all_names (Binding.name "induct")

      in

        ((thm_name, []), [tuple_induct_thm])

      end

    fun unfold_note (name, thm) : Attrib.binding * Thm.thm list =

      let

        val thm_name = Binding.qualify true name (Binding.name "unfold")

        val src = Attrib.internal (K add_unfold)

      in

        ((thm_name, [src]), [thm])

      end

    val (_, lthy) = lthy

      |> fold_map Local_Theory.note (induct_note :: map unfold_note unfold_thms)

  in

    (lthy, names, fixdef_thms, map snd unfold_thms)

  end

(*************************************************************************)

(*********** monadic notation and pattern matching compilation ***********)

(*************************************************************************)

structure FixrecMatchData = Theory_Data

(

  type T = string Symtab.table

  val empty = Symtab.empty

  val extend = I

  fun merge data = Symtab.merge (K true) data

)

(* associate match functions with pattern constants *)

fun add_matchers ms = FixrecMatchData.map (fold Symtab.update ms)

fun taken_names (t : term) : bstring list =

  let

    fun taken (Const(a,_), bs) = insert (op =) (Long_Name.base_name a) bs

      | taken (Free(a,_) , bs) = insert (op =) a bs

      | taken (f $ u     , bs) = taken (f, taken (u, bs))

      | taken (Abs(a,_,t), bs) = taken (t, insert (op =) a bs)

      | taken (_         , bs) = bs

  in

    taken (t, [])

  end

(* builds a monadic term for matching a pattern *)

(* returns (rhs, free variable, used varnames) *)

fun compile_pat match_name pat rhs taken =

  let

    fun comp_pat p rhs taken =

      if is_Free p then (rhs, p, taken)

      else comp_con (fastype_of p) p rhs [] taken

    (* compiles a monadic term for a constructor pattern *)

    and comp_con T p rhs vs taken =

      case p of

        Const(@{const_name Rep_cfun},_) $ f $ x =>

          let val (rhs', v, taken') = comp_pat x rhs taken

          in comp_con T f rhs' (v::vs) taken' end

      | f $ x =>

          let val (rhs', v, taken') = comp_pat x rhs taken

          in comp_con T f rhs' (v::vs) taken' end

      | Const (c, cT) =>

          let

            val n = singleton (Name.variant_list taken) "v"

            val v = Free(n, T)

            val m = Const(match_name c, matcherT (cT, fastype_of rhs))

            val k = big_lambdas vs rhs

          in

            (m`v`k, v, n::taken)

          end

      | _ => raise TERM ("fixrec: invalid pattern ", [p])

  in

    comp_pat pat rhs taken

  end

(* builds a monadic term for matching a function definition pattern *)

(* returns (constant, (vars, matcher)) *)

fun compile_lhs match_name pat rhs vs taken =

  case pat of

    Const(@{const_name Rep_cfun}, _) $ f $ x =>

      let val (rhs', v, taken') = compile_pat match_name x rhs taken

      in compile_lhs match_name f rhs' (v::vs) taken' end

  | Free(_,_) => (pat, (vs, rhs))

  | Const(_,_) => (pat, (vs, rhs))

  | _ => fixrec_err ("invalid function pattern: "

                    ^ ML_Syntax.print_term pat)

fun strip_alls t =

  (case try Logic.dest_all t of

    SOME (_, u) => strip_alls u

  | NONE => t)

fun compile_eq match_name eq =

  let

    val (lhs,rhs) = dest_eqs (Logic.strip_imp_concl (strip_alls eq))

  in

    compile_lhs match_name lhs (mk_succeed rhs) [] (taken_names eq)

  end

(* this is the pattern-matching compiler function *)

fun compile_eqs match_name eqs =

  let

    val (consts, matchers) =

      ListPair.unzip (map (compile_eq match_name) eqs)

    val const =

        case distinct (op =) consts of

          [n] => n

        | [] => fixrec_err "no defining equations for function"

        | _ => fixrec_err "all equations in block must define the same function"

    val vars =

        case distinct (op = o pairself length) (map fst matchers) of

          [vars] => vars

        | _ => fixrec_err "all equations in block must have the same arity"

    (* rename so all matchers use same free variables *)

    fun rename (vs, t) = Term.subst_free (filter_out (op =) (vs ~~ vars)) t

    val rhs = big_lambdas vars (mk_run (foldr1 mk_mplus (map rename matchers)))

  in

    mk_trp (const === rhs)

  end

(*************************************************************************)

(********************** Proving associated theorems **********************)

(*************************************************************************)

fun eta_tac i = CONVERSION Thm.eta_conversion i

fun fixrec_simp_tac ctxt =

  let

    val tab = FixrecUnfoldData.get (Context.Proof ctxt)

    val ss = Simplifier.simpset_of ctxt

    val concl = HOLogic.dest_Trueprop o Logic.strip_imp_concl o strip_alls

    fun tac (t, i) =

      let

        val (c, _) =

            (dest_Const o head_of o chead_of o fst o HOLogic.dest_eq o concl) t

        val unfold_thm = the (Symtab.lookup tab c)

        val rule = unfold_thm RS @{thm ssubst_lhs}

      in

        CHANGED (rtac rule i THEN eta_tac i THEN asm_simp_tac ss i)

      end

  in

    SUBGOAL (fn ti => the_default no_tac (try tac ti))

  end

(* proves a block of pattern matching equations as theorems, using unfold *)

fun make_simps ctxt (unfold_thm, eqns : (Attrib.binding * term) list) =

  let

    val ss = Simplifier.simpset_of ctxt

    val rule = unfold_thm RS @{thm ssubst_lhs}

    val tac = rtac rule 1 THEN eta_tac 1 THEN asm_simp_tac ss 1

    fun prove_term t = Goal.prove ctxt [] [] t (K tac)

    fun prove_eqn (bind, eqn_t) = (bind, prove_term eqn_t)

  in

    map prove_eqn eqns

  end

(*************************************************************************)

(************************* Main fixrec function **************************)

(*************************************************************************)

local

(* code adapted from HOL/Tools/Datatype/primrec.ML *)

fun gen_fixrec

  prep_spec

  (raw_fixes : (binding * 'a option * mixfix) list)

  (raw_spec' : (bool * (Attrib.binding * 'b)) list)

  (lthy : local_theory) =

  let

    val (skips, raw_spec) = ListPair.unzip raw_spec'

    val (fixes : ((binding * typ) * mixfix) list,

         spec : (Attrib.binding * term) list) =

          fst (prep_spec raw_fixes raw_spec lthy)

    val names = map (Binding.name_of o fst o fst) fixes

    fun check_head name =

        member (op =) names name orelse

        fixrec_err ("Illegal equation head. Expected " ^ commas_quote names)

    val chead_of_spec =

      chead_of o fst o dest_eqs o Logic.strip_imp_concl o strip_alls o snd

    fun name_of (Free (n, _)) = tap check_head n

      | name_of _ = fixrec_err ("unknown term")

    val all_names = map (name_of o chead_of_spec) spec

    fun block_of_name n =

      map_filter

        (fn (m,eq) => if m = n then SOME eq else NONE)

        (all_names ~~ (spec ~~ skips))

    val blocks = map block_of_name names

    val matcher_tab = FixrecMatchData.get (Proof_Context.theory_of lthy)

    fun match_name c =

      case Symtab.lookup matcher_tab c of SOME m => m

        | NONE => fixrec_err ("unknown pattern constructor: " ^ c)

    val matches = map (compile_eqs match_name) (map (map (snd o fst)) blocks)

    val spec' = map (pair Attrib.empty_binding) matches

    val (lthy, _, _, unfold_thms) =

      add_fixdefs fixes spec' lthy

    val blocks' = map (map fst o filter_out snd) blocks

    val simps : (Attrib.binding * thm) list list =

      map (make_simps lthy) (unfold_thms ~~ blocks')

    fun mk_bind n : Attrib.binding =

     (Binding.qualify true n (Binding.name "simps"), @{attributes [simp]})

    val simps1 : (Attrib.binding * thm list) list =

      map (fn (n,xs) => (mk_bind n, map snd xs)) (names ~~ simps)

    val simps2 : (Attrib.binding * thm list) list =

      map (apsnd (fn thm => [thm])) (flat simps)

    val (_, lthy) = lthy

      |> fold_map Local_Theory.note (simps1 @ simps2)

  in

    lthy

  end

in

val add_fixrec = gen_fixrec Specification.check_spec

val add_fixrec_cmd = gen_fixrec Specification.read_spec

end (* local *)

(*************************************************************************)

(******************************** Parsers ********************************)

(*************************************************************************)

val opt_thm_name' : (bool * Attrib.binding) parser =

  @{keyword "("} -- @{keyword "unchecked"} -- @{keyword ")"} >> K (true, Attrib.empty_binding)

    || Parse_Spec.opt_thm_name ":" >> pair false

val spec' : (bool * (Attrib.binding * string)) parser =

  opt_thm_name' -- Parse.prop >> (fn ((a, b), c) => (a, (b, c)))

val alt_specs' : (bool * (Attrib.binding * string)) list parser =

  let val unexpected = Scan.ahead (Parse.name || @{keyword "["} || @{keyword "("})

  in Parse.enum1 "|" (spec' --| Scan.option (unexpected -- Parse.!!! (@{keyword "|"}))) end

val _ =

  Outer_Syntax.local_theory "fixrec" "define recursive functions (HOLCF)" Keyword.thy_decl

    (Parse.fixes -- (Parse.where_ |-- Parse.!!! alt_specs')

      >> (fn (fixes, specs) => add_fixrec_cmd fixes specs))

val setup =

  Method.setup @{binding fixrec_simp}

    (Scan.succeed (SIMPLE_METHOD' o fixrec_simp_tac))

    "pattern prover for fixrec constants"

end